Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed. Under this exogenous-x approach, missing covariates are handled primarily via listwise deletion. This sacrifices efficiency and does not allow missingness to depend on observed outcomes. Here we describe a modified joint likelihood approach that (a) allows inference about parameters of the exogenous-x conditional mixture even with nonnormal covariates, unlike a conventional multivariate mixture; (b) retains all cases under missing at random assumptions; (c) yields lower bias and higher efficiency than the exogenous-x approach under a variety of conditions with missing covariates; and (d) is straightforward to implement in available commercial software. The proposed approach is illustrated with an empirical analysis predicting membership in latent classes of conduct problems. Recommendations for practice are discussed.     

Identifying Variables Responsible for Data not Missing at Random

When data are not missing at random (NMAR), maximum likelihood (ML) procedure will not generate consistent parameter estimates unless the missing data mechanism is correctly modeled. Understanding NMAR mechanism in a data set would allow one to better use the ML methodology. A survey or questionnaire may contain many items; certain items may be responsible for NMAR values in other items. The paper develops statistical procedures to identify the responsible items. By comparing ML estimates (MLE), statistics are developed to test whether the MLEs are changed when excluding items. The items that cause a significant change of the MLEs are responsible for the NMAR mechanism. Normal distribution is used for obtaining the MLEs; a sandwich-type covariance matrix is used to account for distribution violations. The class of nonnormal distributions within which the procedure is valid is provided. Both saturated and structural models are considered. Effect sizes are also defined and studied. The results indicate that more missing data in a sample does not necessarily imply more significant test statistics due to smaller effect sizes. Knowing the true population means and covariances or the parameter values in structural equation models may not make things easier either. The research was supported by NSF grant DMS04-37167, the James McKeen Cattell Fund.     

Estimating Incremental Validity Under Missing Data

A common form of missing data is caused by selection on an observed variable (e.g., Z). If the selection variable was measured and is available, the data are regarded as missing at random (MAR). Selection biases correlation, reliability, and effect size estimates when these estimates are computed on listwise deleted (LD) data sets. On the other hand, maximum likelihood (ML) estimates are generally unbiased and outperform LD in most situations, at least when the data are MAR. The exception is when we estimate the partial correlation. In this situation, LD estimates are unbiased when the cause of missingness is partialled out. In other words, there is no advantage of ML estimates over LD estimates in this situation. We demonstrate that under a MAR condition, even ML estimates may become biased, depending on how partial correlations are computed. Finally, we conclude with recommendations about how future researchers might estimate partial correlations even when the cause of missingness is unknown and, perhaps, unknowable.     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

Comparison of Modern Methods for Analyzing Repeated Measures Data With Missing Values

Missing data are a pervasive problem in many psychological applications in the real world. In this article we study the impact of dropout on the operational characteristics of several approaches that can be easily implemented with commercially available software. These approaches include the covariance pattern model based on an unstructured covariance matrix (CPM-U) and the true covariance matrix (CPM-T), multiple imputation-based generalized estimating equations (MI-GEE), and weighted generalized estimating equations (WGEE). Under the missing at random mechanism, the MI-GEE approach was always robust. The CPM-T and CPM-U methods were also able to control the error rates provided that certain minimum sample size requirements were met, whereas the WGEE was more prone to inflated error rates. In contrast, under the missing not at random mechanism, all evaluated approaches were generally invalid. Our results also indicate that the CPM methods were more powerful than the MI-GEE and WGEE methods and their superiority was often substantial. Furthermore, we note that little or no power was sacrificed by using CPM-U method in place of CPM-T, although both methods have less power in situations where some participants have incomplete data. Some aspects of the CPM-U and MI-GEE methods are illustrated using real data from 2 previously published data sets. The first data set comes from a randomized study of AIDS patients with advanced immune suppression, the second from a cohort of patients with schizotypal personality disorder enrolled in a prevention program for psychosis.     

Covariance Structure Model Fit Testing Under Missing Data: An Application of the Supplemented EM Algorithm

Analysis of covariance or correlation structure is characterized, unfortunately, by repetitive suboptimal communication between persons and/or computer programs. After analyzing some aspects of this suboptimality, I suggest some approaches to improving the situation.     

基于增长模型的非随机缺失数据处理:选择模型和极大似然方法

对含有非随机缺失数据的潜变量增长模型,为了考察基于不同假设的缺失数据处理方法:极大似然(ML)方法与DiggleKenward选择模型的优劣,通过Monte Carlo模拟研究,比较两种方法对模型中增长参数估计精度及其标准误估计的差异,并考虑样本量、非随机缺失比例和随机缺失比例的影响。结果表明,符合前提假设的Diggle-Kenward选择模型的参数估计精度普遍高于ML方法;对于标准误估计值,ML方法存在一定程度的低估,得到的置信区间覆盖比率也明显低于Diggle-Kenward选择模型。     

Not fixating at the line of text comes at a cost

Previous research on eye guidance in reading has investigated systematic tendencies with respect to horizontal fixation locations on letters within words and the relationship between fixation location in a word and the duration of the fixation. The present study investigates where readers place their eyes vertically on the line of text and how vertical fixation location is related to fixation duration. Analyses were based on a large corpus of eye movement recordings from single-sentence reading. The vertical preferred viewing location was found to be within the vertical extent of the font, but fixations beyond the vertical boundaries of the text also frequently occurred. Analyzing fixation duration as a function of vertical fixation location revealed a vertical optimal viewing position (vOVP) effect: Fixations were shortest when placed optimally on the line of text, and fixation duration gradually increased for fixations that fell above or below the line of text. The vOVP effect can be explained by the limits of visual resolution along the vertical meridian. It is concluded that vertical and horizontal landing positions in single-sentence reading are associated with differences in fixation durations in opposite ways.     

Evaluating Word-Reading Models at the Item Level: Matching the Grain of Theory and Data

Spieler and Balota (1997) showed that connectionist models of reading account for relatively little item-specific variance. In assessing this finding, it is important to recognize two factors that limit how much variance such models can possibly explain. First, item means are affected by several factors that are not addressed in existing models, including processes involved in recognizing letters and producing articulatory output. These limitations point to important areas for future research but have little bearing on existing theoretical claims. Second, the item data include a substantial amount of error variance that would be inappropriate to model. Issues concerning comparisons between simulation data and human performance are discussed with an emphasis on the importance of evaluating models at a level of specificity ("grain") appropriate to the theoretical issues being addressed.     

The Performance of Methods to Test Upper-Level Mediation in the Presence of Nonnormal Data

A Monte Carlo study compared the statistical performance of standard and robust multilevel mediation analysis methods to test indirect effects for a cluster randomized experimental design under various departures from normality. The performance of these methods was examined for an upper-level mediation process, where the indirect effect is a fixed effect and a group-implemented treatment is hypothesized to impact a person-level outcome via a person-level mediator. Two methods—the bias-corrected parametric percentile bootstrap and the empirical-M test—had the best overall performance. Methods designed for nonnormal score distributions exhibited elevated Type I error rates and poorer confidence interval coverage under some conditions. Although preliminary, the findings suggest that new mediation analysis methods may provide for robust tests of indirect effects.     

Training for Task Performance Under Stress: The Effectiveness of Phased Training Methods

Studies on the training of individuals for task performance in stressful situations have typically considered two approaches. One approach requires that, while training on the task, trainees be exposed to stressors of a kind and intensity characteristic of the situations for which they are being trained ("high fidelity" training). Such an approach might suffer from the interference of stressors with task acquisition. Another approach allows the trainee to train on the task in a stress-free environment or under low-intensity stressors ("low fidelity" training). This approach leaves the trainee insufficiently prepared for task performance under stress. The present study compared these two basic approaches to three forms of "phased" training, which consisted of different combinations of three separate and distinct training phases: a phase which allows the trainee to acquire the task under stress- free conditions; a phase which allows him or her to passively experience the stressor; and a phase in which newly acquired skills are practiced under stress. The results showed that a phased training process which combines the first and third phases just described, is more effective than either "high fidelity" training or "low fidelity" training.     

Postmodeling Sensitivity Analysis to Detect the Effect of Missing Data Mechanisms

Incomplete or missing data is a common problem in almost all areas of empirical research. It is well known that simple and ad hoc methods such as complete case analysis or mean imputation can lead to biased and/or inefficient estimates. The method of maximum likelihood works well; however, when the missing data mechanism is not one of missing completely at random (MCAR) or missing at random (MAR), it too can result in incorrect inference. Statistical tests for MCAR have been proposed, but these are restricted to a certain class of problems. The idea of sensitivity analysis as a means to detect the missing data mechanism has been proposed in the statistics literature in conjunction with selection models where conjointly the data and missing data mechanism are modeled. Our approach is different here in that we do not model the missing data mechanism but use the data at hand to examine the sensitivity of a given model to the missing data mechanism. Our methodology is meant to raise a flag for researchers when the assumptions of MCAR (or MAR) do not hold. To our knowledge, no specific proposal for sensitivity analysis has been set forth in the area of structural equation models (SEM). This article gives a specific method for performing postmodeling sensitivity analysis using a statistical test and graphs. A simulation study is performed to assess the methodology in the context of structural equation models. This study shows success of the method, especially when the sample size is 300 or more and the percentage of missing data is 20% or more. The method is also used to study a set of real data measuring physical and social self-concepts in 463 Nigerian adolescents using a factor analysis model.     

On the Model-Based Bootstrap With Missing Data: Obtaining a P-Value for a Test of Exact Fit

Evaluating the fit of a structural equation model via bootstrap requires a transformation of the data so that the null hypothesis holds exactly in the sample. For complete data, such a transformation was proposed by Beran and Srivastava (1985) Beran, R. and Srivastava, M. S. 1985. Bootstrap tests and confidence regions for functions of a covariance matrix. The Annals of Statistics, 13: 95–115. [Crossref], [Web of Science ®] , [Google Scholar] for general covariance structure models and applied to structural equation modeling by Bollen and Stine (1992) Bollen, K. A. and Stine, R. A. 1992. Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods and Research, 21: 205–229. [Crossref], [Web of Science ®] , [Google Scholar]. An extension of this transformation to missing data was presented by Enders (2002) Enders, C. K. 2002. Applying the Bollen-Stine bootstrap for goodness-of-fit measures to structural equation models with missing data. Multivariate Behavioral Research, 37: 359–377. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], but it is an approximate and not an exact solution, with the degree of approximation unknown. In this article, we provide several approaches to obtaining an exact solution. First, an explicit solution for the special case when the sample covariance matrix within each missing data pattern is invertible is given. Second, 2 iterative algorithms are described for obtaining an exact solution in the general case. We evaluate the rejection rates of the bootstrapped likelihood ratio statistic obtained via the new procedures in a Monte Carlo study. Our main finding is that model-based bootstrap with incomplete data performs quite well across a variety of distributional conditions, missing data mechanisms, and proportions of missing data. We illustrate our new procedures using empirical data on 26 cognitive ability measures in junior high students, published in Holzinger and Swineford (1939) Holzinger, K. J. and Swineford, F. 1939. A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, 48: 1–91.  [Google Scholar].     

Treatment of Missing Data: Beyond Ends and Means

The ethical decision making process behind the treatment of missing data has yet to be examined in the research literature in any discipline. The purpose of the current paper is to begin to discuss this decision-making process in view of a Foucauldian framework. The paper suggests how the ethical treatment of missing data should be considered from the adoption of this theoretical framework.     

概化理论缺失数据方差分量估计

各种心理调查、心理实验中, 数据的缺失随处可见。由于数据缺失, 给概化理论分析非平衡数据的方差分量带来一系列问题。基于概化理论框架下, 运用Matlab 7.0软件, 自编程序模拟产生随机双面交叉设计p×i×r缺失数据, 比较和探讨公式法、REML法、拆分法和MCMC法在估计各个方差分量上的性能优劣。结果表明：(1) MCMC方法估计随机双面交叉设计p×i×r缺失数据方差分量, 较其它3种方法表现出更强的优势; (2) 题目和评分者是缺失数据方差分量估计重要的影响因素。